140 CHAPTER 5. THE DERIVATIVE

7. Show that, like continuous functions, functions which are derivatives have the in-termediate value property. This means that if f ′ (a) < 0 < f ′ (b) then there existsx ∈ (a,b) such that f ′ (x) = 0. Hint: Argue the minimum value of f occurs at aninterior point of [a,b] .

8. Find an example of a function which has a derivative at every point but such thatthe derivative is not everywhere continuous. Hint: Consider something involvingx2 sin(1/x).

9. ∗Let f be a real continuous function defined on the interval [0,1] . Also supposef (0) = 0 and f (1) = 1 and f ′ (t) exists for all t ∈ (0,1) . Show there exists n distinctpoints {si}n

i=1 of the interval such that ∑ni=1 f ′ (si) = n. Hint: Consider the mean

value theorem applied to successive pairs in the following sum. f( 1

3

)− f (0) +

f( 2

3

)− f

( 13

)+ f (1)− f

( 23

)10. ∗Now suppose f : [0,1]→ R is continuous and differentiable on (0,1) and f (0) = 0

while f (1) = 1. Show there are distinct points {si}ni=1 ⊆ (0,1) such that

n

∑i=1

(f ′ (si)

)−1= n.

Hint: Let 0 = t0 < t1 < · · · < tn = 1 and pick xi ∈ f−1 (ti) such that these xi areincreasing and xn = 1,x0 = 0. Explain why you can do this. Then argue ti+1 − ti =f (xi+1)− f (xi) = f ′ (si)(xi+1 − xi) and so xi+1−xi

ti+1−ti= 1

f ′(si). Now choose the ti to be

equally spaced.

11. Show that (x+1)3/2 − x3/2 > 2 for all x ≥ 2. Explain why for n a natural numberlarger than or equal to 1, there exists a natural number m such that (n+1)3 >m2 > n3.Hint: Verify directly for n = 1 and use the above inequality to take care of the casewhere n ≥ 2. This shows that between the cubes of any two natural numbers there isthe square of a natural number. This interesting fact was used by Jacobi in 1835 toshow a very important theorem in complex analysis.

12. An initial value problem for undamped vibration is

differential equation︷ ︸︸ ︷y′′+ω

2y = 0 ,

initial conditions︷ ︸︸ ︷y(0) = y0,y′ (0) = y1

You are looking for a function y(t) which satisfies this equation.

(a) First show that if you have a complex valued function z(t) satisfying the differ-ential equation, then the real and imaginary parts of z denoted by Rez and Imzalso solve the differential equation.

(b) Show that if y1 and y2 solve the differential equation, then if C1,C2 are arbitraryconstants, then C1y1 +C2y2 also solves the differential equation.

(c) Now use Euler’s formula in Section 5.6 to show that z= eiωt ,z= e−iωt solve thedifferential equation. Use the first part to find that y1 (t) = sinωt and y2 (t) =cosωt both solve the above equation.

14010.11.12.CHAPTER 5. THE DERIVATIVEShow that, like continuous functions, functions which are derivatives have the in-termediate value property. This means that if f’(a) < 0 < f’(b) then there existsx € (a,b) such that f’ (x) = 0. Hint: Argue the minimum value of f occurs at aninterior point of [a,b].Find an example of a function which has a derivative at every point but such thatthe derivative is not everywhere continuous. Hint: Consider something involvingx? sin (1/x).*Let f be a real continuous function defined on the interval [0,1]. Also supposef (0) =O and f (1) = 1 and f’ (t) exists for all t € (0,1). Show there exists n distinctpoints {s;};_, of the interval such that Y!_, f’ (s;) =n. Hint: Consider the meanvalue theorem applied to successive pairs in the following sum. f (4) — f (0) +FG) -F(G) +4) -£(3)*Now suppose f : [0,1] + R is continuous and differentiable on (0,1) and f (0) =0while f (1) = 1. Show there are distinct points {s;}/_, C (0,1) such thatHint: Let 0 =f <t < ++: <t, = 1 and pick x; € f~!(t;) such that these x; areincreasing and x, = 1,x9 = 0. Explain why you can do this. Then argue ¢j41 —t; =f (xi4) — f (xi) = f’ (si) (x41 — 2x7) and so aT = Fay: Now choose the ¢; to beequally spaced.Show that (x + 1)°/ 2 _ 3/2 5 2 for all x > 2. Explain why for 7 a natural numberlarger than or equal to 1, there exists a natural number m such that (n + 1° >m > n>.Hint: Verify directly for n = 1 and use the above inequality to take care of the casewhere n > 2. This shows that between the cubes of any two natural numbers there isthe square of a natural number. This interesting fact was used by Jacobi in 1835 toshow a very important theorem in complex analysis.An initial value problem for undamped vibration isdifferential equation initial conditionsy"+@°y=0 ,y(0) =o,’ (0) =y1You are looking for a function y(t) which satisfies this equation.(a) First show that if you have a complex valued function z(t) satisfying the differ-ential equation, then the real and imaginary parts of z denoted by Rez and Imzalso solve the differential equation.(b) Show that if y; and y2 solve the differential equation, then if C; ,C2 are arbitraryconstants, then C)y; + C2y2 also solves the differential equation.(c) Now use Euler’s formula in Section 5.6 to show that z= e',z =e solve thedifferential equation. Use the first part to find that y; (t) = sin? and yo (t) =cos ot both solve the above equation.